An Inverse Source Problem For A Time-space Fractional Diffusion Equation

In recent decades,the inverse problem have been appeared in many fields such as life sciences,image processing,and remote sensing technology.Due to the applicability of inverse problems,the research on inverse problems has been rapidly developed in the field of applied mathematics.Therefore,the study of the inverse problem is necessary.In this paper,we consider the inverse source problem for the space source term in the time-space fractional diffusion equations.In the first chapter,we introduce the background and development of the fractional diffusion equation,state the historical background of the inverse problem and the ill-posed problem,and trace the results obtained by the regularization method.Then,a brief description of the work to be studied in this paper is outlined,and the basic definitions and lemmas needed in this paper are listed.In chapter 2,we study a class of diffusion equations with left Caputo fractional derivative and space fractional Laplace operator.By means of the method of separation of variables,the analytical expression of the forward problem of the this kind of equation is discussed.Firstly,the space source term f(x)in the time-space fractional diffusion equation is determined by analyzing the initial boundary data and the value u(x,T)=g(x)at the time of final state.Secondly,combining with the analytical expression of the forward problem,the inverse problem of the space source term is transformed into the solution of the first Fredholm integral equation.It is proved that the solution of the equation cannot continuously depend on the given data,that is,the integral equation is an ill-posed problem.On the basis of the above work,we discuss the uniqueness of the solution of the inverse source problem.In chapter 3,in order to solve the ill-posed problem of inverse source,Tikhonov regularization method is used to obtain the regular solution of this problem.On the one hand,using the prior regularization parameter selection rule,the regularization parameters are selected according to the prior conditions and noise estimation,and then the convergence rate of the regularization solution of the inverse source problem is obtained.On the other hand,using the selection rules of posterior regularization parameters and combining with the Morozov deviation principle in this paper,we prove the convergence of the regular solution of the inverse source problem.The results in this paper complement and improve the relevant conclusions of the existing literatures.In the fourth chapter,a numerical case is given to prove the effectiveness and feasibility of the above method which under the posterior regularization parameter selection rule.Finally,our research works and the prospect of our future research are summarized.